# Small-Signal Analysis of V^{2} Control Using Equivalent Circuit Model of Current Mode Controls

^{2}control, the direct feedback contains the information of both state variables. In this paper, by separating current feedback and capacitor voltage feedback, an equivalent circuit of V

^{2}control based on the equivalent circuit of current mode control is proposed. The proposed equivalent circuit provides a clear physical insight of V

^{2}control and can be interpreted as an advanced implementation of current mode control with a proportional voltage feedback, as well as an additional load current feedback. The load current feed-forward dramatically reduces the output impedance. The model is extended to enhanced V

^{2}control. The proposed equivalent circuit model is applicable to both variable frequency modulation and constant frequency modulation.

The direct output voltage V

_{o}feedback consists of inductor current, capacitor voltage and load current feedback. The inductor current feedback loop does not have a low pass filter, so all the sideband frequencies (f

_{sw}-f

_{m}, f

_{sw}+f

_{m}, etc.) are fed back to the modulator. Other feedbacks are simpler: for a voltage regulator, the impedance of the capacitor branch is much smaller than that of the load resistor at f >f

_{sw}/10. Assuming the capacitor ESR zero is well below half switching frequency, the impedance of ESR is much larger than that of intrinsic capacitance, so the sideband components at v

_{cap}are overwhelmed by the ESR voltage.

According to the analysis above, it is reasonable to consider all the sideband frequency feedback effect in the inductor current loop, but only consider fundamental frequency in the capacitor voltage and load current feedback. Inductor current loop is a highly nonlinear entity. It potentially has sub-harmonic instability. A three-terminal switch equivalent circuit model based on the result of a describing function derivation accurately predicts the small signal properties of this nonlinear entity. Substitute the closed current loop and the PWM switch by the equivalent circuit mode. The proposed equivalent circuit model of V

^{2}control is shown in Fig. 1 and the simulation verification of the proposed model is shown in Fig. 2.